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bryan k differential equations toolbox modeling 2021

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Textbook in PDF format Foreword Preface Why Study Differential Equations? The Olympic -Meter Dash Usain Bolt's Olympic Victory Modeling a Sprint The Hill-Keller Differential Equation Intracochlear Drug Delivery The Challenge of Hearing Loss A Compartmental Model for the Cochlea The Differential Equation Population Growth and Fishery Management The Need to Manage Fish Harvesting Modeling Fish Population Modeling Harvesting Parameter Estimation and Harvesting Where Do We Go from Here? A Toolbox for Describing the World Some Terminology You Already Know How to Solve Some Differential Equations Exercises The Blessing of Dimensionality Definition of Dimension The Algebra of Dimension Derivatives, Integrals, Elementary Functions Unit-Free Equations and Bending the Rules Using Dimension to Find Plausible Models Other Dimensions Exercises Modeling Projects Project: Hang Time Project: Money Matters Project: Ant Tunneling First-Order Equations First-Order Linear Equations Example: Solving the Hill-Keller Equation as a Linear ODE A General Procedure for Solving Linear ODEs Some Common First-Order Linear Models Exercises Separable Equations Application: Falling Objects Separation of Variables: A First Example The General Procedure for Separation of Variables Example: Solving the Falling Object ODE Example: Solving the Logistic Equation Exercises Qualitative and Graphical Insights Direction Fields Autonomous Equations Phase Portraits Fixed Points and Stability Determining the Stability of Fixed Points Bifurcations Exercises The Existence and Uniqueness of Solutions Some Inspiration from Calculus What Are Solutions to ODEs? The Existence-Uniqueness Theorem for ODEs Exercises Modeling Projects Project: Money Matters Project: Chemical Kinetics Project: A Shot in the Water Numerical Methods for ODEs The Need for Numerics Logistic Example: Time-Varying Parameters Euler's Method Evaluate, Extrapolate, Repeat as Necessary The Accuracy of Euler's Method Exercises Improvements to Euler's Method Improving Euler's Method The Improved Euler Method Exercises Modern Numerical Methods The RK Algorithm Adaptive Step Sizing and Error Control Exercises Parameter Estimation Hill-Keller Revisited Least-Squares Estimation Hill-Keller Again Least Squares For ODE Parameter Estimation A Cautionary Example Exercises Modeling Projects Project: Sublimation of Carbon Dioxide Project: Fish Harvesting Revisited Project: The Mathematics of Marriage Project: Shuttlecocks and the Akaike Information Criterion Second-Order Equations Vibration and the Harmonic Oscillator The Chilean Earthquake The Harmonic Oscillator Initial Conditions More Applications of Spring-Mass Models Exercises The Harmonic Oscillator Solving the Harmonic Oscillator ODE: Examples Solving Second-Order Linear ODEs: The General Case The Underdamped and Undamped Cases The General Underdamped Case The Critically Damped Case The Existence and Uniqueness of Solutions Summary and a Physical Perspective Exercises The Forced Harmonic Oscillator Solving the Forced Harmonic Oscillator Equation Finding a Particular Solution: Undetermined Coefficients When the Guess Fails Exercises Resonance An Example of Resonance Periodic Forcing Exercises Scaling and Nondimensionalization for ODEs Motivation: Nonlinear Springs Characteristic Variable Scales Nondimensionalization: Logistic Equation Example Nondimensionalization: Harvested Logistic Equation Example The General Outline for Nondimensional Rescaling Back to the Hard Spring Exercises Modeling Projects Project: Earthquake Modeling Project: Stay Tuned—RLC Circuits and Radios Project: Parameter Estimation with Second-Order ODEs Project: Bike Shock Absorber Project: The Pendulum Project: The Pendulum The Laplace Transform Discontinuous Forcing Functions Motivation: Pharmacokinetics Complication: Discontinuous Forcing Complication: Impulsive Forcing Discontinuous Forcing and Transform Methods Exercises The Laplace Transform Definition of the Laplace Transform What Kinds of Functions Can Be Transformed? Laplace Transforms of Elementary Functions Solving Differential Equations Using Laplace Transforms The First Shifting Theorem The Inverse Laplace Transform The Initial and Final Value Theorems Section Summary and Remarks Exercises Nonhomogeneous Problems and Discontinuous Forcing Functions Some Nonhomogeneous Examples Discontinuous Forcing The Second Shifting Theorem Some More Models and Examples Summary and Remarks Exercises The Dirac Delta Function Motivational Examples Definition of the Dirac Delta Function Three Models: Money, Masses, and Medication The Laplace Transform of the Dirac Delta Function Solving ODEs with Dirac Delta Functions Summary and a Few Remarks Laplace Transform Table Exercises Input-Output, Transfer Functions, and Convolution A System Identification Problem Input-Output Systems Convolution The Impulse Response and Convolution System Identification with Impulsive Input Exercises A Taste of Control Theory The Need for Control Modeling an Incubator Open-Loop Control Closed-Loop Control Proportional-Integral Control Proportional-Integral-Derivative Control Disturbances Summary and Comments Exercises Modeling Projects Project: Drug Dosage Project: Machine Replacement Project: Vibration Isolation Table Shakedown Project: Segway Scooters and The Inverted Pendulum Linear Systems of Differential Equations Systems of Differential Equations Motivation: More Pharmacokinetics Existence and Uniqueness Exercises Linear Constant-Coefficient Homogeneous Systems of ODEs Matrix-Vector Formulation Solving the Homogeneous Case Complex Eigenvalues Defective Matrices Exercises Linear Constant-Coefficient Nonhomogeneous Systems of ODEs Solving Linear Systems of ODEs with Laplace Transforms Undetermined Coefficients for Systems of ODEs The Significance of Eigenvalues Exercises The Matrix Exponential Inspiration Definition of the Matrix Exponential Properties of the Matrix Exponential Solving ODEs with the Matrix Exponential Computing The Matrix Exponential: The Diagonal Case Computing The Matrix Exponential: The Diagonalizable Case Computing The Matrix Exponential: Putzer's Algorithm Final Remarks Exercises Modeling Projects Project: LSD Compartment Model Project: Homelessness Project: Tuned Mass Dampers Nonlinear Systems of Differential Equations Autonomous Nonlinear Systems and Direction Fields Some Nonlinear ODE Models Direction Fields A Nonlinear Direction Field Example Direction Fields in Higher Dimensions Exercises Direction Fields and Phase Portraits for Linear Systems Direction Fields for Homogeneous Linear Systems Application to the LSD Model The Equation =Ax+b Direction Fields for Larger Systems of ODEs Exercises Autonomous Nonlinear Systems and Phase Portraits Sketching Phase Portraits for Nonlinear Systems Linearizing ODEs at Equilibrium Points Exercises Analyzing Systems with Unspecified Parameters Sketching Phase Portraits with Unspecified Parameters Linearizing the Competing Species Model with General Parameters Conclusions for Competing Species Higher-Dimensional Systems Exercises Numerical Methods for Systems of First Order ODE's Extending Basic Numerical Methods to Systems Stiff Systems of ODEs Implicit Numerical ODE Solvers Exercises Additional Techniques for Systems of First Order ODEs First Integrals and Conservative Systems Lyapunov Functions Linearization and the Routh-Hurwitz Theorem Exercises Modeling Projects Project: Homelessness Revisited Project: Predator-Prey Model Project: Parameter Estimation for Competing Yeast Species An Introduction to Partial Differential Equations Conservation of Stuff and the Continuity Equation Industrial Furnaces and Metal Production Conservation of Stuff The Continuity Equation The Heat Equation Some Solutions to the Heat Equation: Separation of Variables and Linearity Exercises Fourier Series An Example Approximating Functions The Fourier Cosine Expansion The Fourier Sine Expansion More on Fourier Series Convergence Exercises Solving the Heat Equation Homogeneous Dirichlet Conditions Insulating Boundary Conditions Other Boundary Conditions Diffusion Solving the Nonhomogeneous Heat or Diffusion Equation Exercises The Advection and Wave Equations The Advection Equation Solution to the Advection Equation The Wave Equation Solution to the Wave Equation The Wave Equation on the Real Line Exercises Modeling Projects Project: It's a Blast (Furnace)! Project: Finding Polluters Project: Strung Out Project: Frequency Analysis of Signals Project: It's All Relative Appendix A Complex Numbers Motivation and Definition Arithmetic with Complex Numbers Exponentiation of Complex Numbers The Fundamental Theorem of Algebra Partial Fraction Decompositions over the Complex Numbers Additional Exercises Appendix B Matrix Algebra Linear System of Equations Matrix Algebra Eigenvalues and Eigenvectors The Eigenvalues for a General Two by Two Matrix Diagonalization Additional Exercises Appendix C Circuits Current, Voltage, and Resistance Capacitors Inductors RLC Circuits Complex-Valued Solutions and Periodic Forcing Impedance in Electrical Circuits Bibliography Index Back Cover

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